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CHAPTER 27 Quantum Physics

Units

• Discovery and Properties of the Electron • Planck’s Quantum Hypothesis; Blackbody Radiation • Photon Theory of Light and the Photoelectric Effect • Energy, Mass, and Momentum of a Photon • Compton Effect • Photon Interactions; Pair Production • Wave-Particle Duality; the Principle of Complementarity • Wave Nature of Matter • Electron Microscopes • Early Models of the Atom • Atomic Spectra: Key to the Structure of the Atom • The Bohr Model • de Broglie’s Hypothesis Applied to Atoms

Need for Quantum Physics

• Problems remained from classical mechanics that relativity didn’t explain • Blackbody Radiation

– The electromagnetic radiation emitted by a heated object • Photoelectric Effect

– Emission of electrons by an illuminated metal • Spectral Lines

– Emission of sharp spectral lines by gas atoms in an electric discharge tube

Development of Quantum Physics

• 1900 to 1930 – Development of ideas of quantum mechanics

• Also called wave mechanics • Highly successful in explaining the behavior of atoms, molecules, and

nuclei • Involved a large number of physicists

– Planck introduced basic ideas – Mathematical developments and interpretations involved such people as Einstein,

Bohr, Schrödinger, de Broglie, Heisenberg, Born and Dirac

Blackbody Radiation

• An object at any temperature emits electromagnetic radiation – Sometimes called thermal radiation – Stefan’s Law describes the total power radiated

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– The spectrum of the radiation depends on the temperature and properties of the object

Blackbody Radiation and Planck’s Hypothesis of Quantized Energy

An ideal blackbody absorbs all the light that is incident upon it.

• Experimental data for distribution of energy in blackbody radiation

• As the temperature increases, the total amount of energy increases

– Shown by the area under the curve • As the temperature increases, the peak of the

distribution shifts to shorter wavelengths

Wien’s Displacement Law

• The wavelength of the peak of the blackbody distribution was found to follow Wein’s Displacement Law

– λmax T = 0.2898 x 10-2 m • K

• λmax is the wavelength at which the curve’s peak

• T is the absolute temperature of the object emitting the radiation

The Ultraviolet Catastrophe

• Classical theory did not match the experimental data • At long wavelengths, the match is good • At short wavelengths, classical theory predicted infinite

energy • At short wavelengths, experiment showed no energy • This contradiction is called the ultraviolet catastrophe

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Max Planck

• 1858 – 1947 • Introduced a “quantum of action,” h • Awarded Nobel Prize in 1918 for discovering the quantized

nature of energy

Planck’s Quantum Hypothesis; Blackbody Radiation

One of the observations that was unexplained at the end of the nineteenth century was the spectrum of light emitted by hot objects. All objects emit radiation whose total intensity is proportional to the fourth power of their temperature. This is called thermal radiation; a blackbody is one that emits thermal radiation only.

The spectrum of blackbody radiation has been measured; it is found that the frequency of peak intensity increases linearly with temperature. This figure shows blackbody radiation curves for three different temperatures. Note that frequency increases to the left.

Planck’s Resolution

• Planck hypothesized that the blackbody radiation was produced by resonators

– Resonators were submicroscopic charged oscillators • The resonators could only have discrete energies

– En = n h ƒ

• n is called the quantum number • ƒ is the frequency of vibration

• h is Planck’s constant, 6.626 x 10-34 J s • Key point is quantized energy states

Example 1: Estimate the temperature of the surface of the Sun, given that the Sun emits light whose peak intensity occurs in the visible spectrum at around 500nm.

32.90 10pT x m K

3 3

9

2.90 10 2.90 106000

500 10p

x m K x m KT K

x m

4

Example 2: Suppose a star has a surface temperature of 32,500K. What color would this star appear?

The peak is in the UV range of the spectrum. The star will appear bluish (or blue-white).

Example 3: A 2.00-kg mass is attached to a spring having force constant k = 25.0N/m and negligible mass. The spring is stretched 0.400 m from its equilibrium position and released. (a) Find the total energy and frequency of oscillation according to classical calculations.

(b) Assume that Plank’s law of energy quantization applies to any oscillator, atomic or large scale, and find the quantum number n for this system.

(c) How much energy would be carried away in a one-quantum change?

Planck’s Quantum Hypothesis; Blackbody Radiation

This spectrum could not be reproduced using 19th-century physics. A solution was proposed by Max Planck in 1900: The energy of atomic oscillations within atoms cannot have an arbitrary value; it is related to the frequency:

The constant h is now called Planck’s constant. 346.626 10h x J s

Photoelectric Effect

• When light is incident on certain metallic surfaces, electrons are emitted from the surface – This is called the photoelectric effect – The emitted electrons are called photoelectrons

3 3

4

2.90 10 2.90 1089.2

3.25 10p

x m K x m Knm

T x K

2 21 1(25.0 / )(0.400 ) 2.00

2 2

1 1 25.0 /0.563

2 2 2.00

E kA N m m J

k N mf Hz

m kg

33

34

2.005.36 10

(6.63 10 )(0.563 )

nn

EE nhf n

hf

Jn x

x J s Hz

1

34 34 (6.63 10 )(0.563 ) 3.73 10

n nE E E hf

x J s Hz x J

5

• The effect was first discovered by Hertz • The successful explanation of the effect was given by Einstein in 1905

– Received Nobel Prize in 1921 for paper on electromagnetic radiation, of which the photoelectric effect was a part

Photoelectric Effect Schematic

• When light strikes E, photoelectrons are emitted • Electrons collected at C and passing through the

ammeter are a current in the circuit • C is maintained at a positive potential by the power

supply

Photon Theory of Light and the Photoelectric Effect

The photoelectric effect: If light strikes a metal, electrons are emitted. The effect does not occur if the frequency of the light is too low; the kinetic energy of the electrons increases with frequency.

If light is a wave, theory predicts: 1. Number of electrons and their energy should increase with intensity 2. Frequency would not matter

If light is a particle, theory predicts: • Increasing intensity increases number of electrons but not energy • Above a minimum energy required to break atomic bond, kinetic energy will increase

linearly with frequency • There is a cutoff frequency below which no electrons will be emitted, regardless of

intensity

The particle theory assumes that an electron absorbs a single photon. Plotting the kinetic energy vs. frequency:

This shows clear agreement with the photon theory, and not with wave theory.

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Features Not Explained by Classical Physics/Wave Theory

• No electrons are emitted if the incident light frequency is below some cutoff frequency that is characteristic of the material being illuminated

• The maximum kinetic energy of the photoelectrons is independent of the light intensity • The maximum kinetic energy of the photoelectrons increases with increasing light

frequency • Electrons are emitted from the surface almost instantaneously, even at low intensities

Einstein’s Explanation

• A tiny packet of light energy, called a photon, would be emitted when a quantized oscillator jumped from one energy level to the next lower one

– Extended Planck’s idea of quantization to electromagnetic radiation • The photon’s energy would be E = hƒ • Each photon can give all its energy to an electron in the metal • The maximum kinetic energy of the liberated photoelectron is KEmax = hƒ – Φ

• Φ is called the work function of the metal

Explanation of Classical “Problems”

• The effect is not observed below a certain cutoff frequency since the photon energy must be greater than or equal to the work function

– Without this, electrons are not emitted, regardless of the intensity of the light • The maximum KE depends only on the frequency and the work function, not on the

intensity • The maximum KE increases with increasing frequency • The effect is instantaneous since there is a one-to-one interaction between the photon and

the electron

Verification of Einstein’s Theory

• Experimental observations of a linear relationship between KE and frequency confirm Einstein’s theory

• The x-intercept is the cutoff frequency • The cutoff wavelength is related to the work

function

chc

• Wavelengths greater than l

C incident on a

material with a work function f don’t result in the emission of photoelectrons

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Photocells

• Photocells are an application of the photoelectric effect • When light of sufficiently high frequency falls on the cell, a current is produced • Examples

– Streetlights, garage door openers, elevators

The photoelectric effect is how “electric eye” detectors work. It is also used for movie film soundtracks.

Example 4: Calculate the energy of a photon of blue light, 450nm in air.

or

Example 5: A sodium surface is illuminated with light of wavelength 0.300 . The work function for

sodium is 2.46eV. (a) Calculate the energy of each photon in electron volts.

(b) Calculate the maximum kinetic energy of the ejected photoelectrons.

(c) Calculate the cutoff wavelength for sodium.

where /E hf f c

34 8

7

(6.63 10 )(3.0 10 / )

(4.5 10 )

hc x J s x m sE hf

x m

194.4 10x J

19 19(4.4 10 ) /(1.60 10 / ) 2.8x J x J eV eV

815

6

3.00 10 /1.00 10

0.300 10

c x m sc f f x Hz

x m

34 15 19(6.63 10 )(1.00 10 ) 6.63 10E hf x J s x Hz x J

19

19

1.00(6.63 10 ) 4.14

1.60 10

eVE x J eV

x J

max 4.14 2.46 1.68KE hf eV eV eV

19 192.46 (2.46 )(1.60 10 / ) 3.94 10eV eV x J eV x J

34 87

19

(6.63 10 )(3.00 10 / )5.05 10

3.94 10c

hc x J s x m sx m

x J

505nm

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Example 6: Estimate how many visible light photons a 100-W lightbulb emits per second. Assume the bulb has a typical efficiency of about 3% (97% of the energy goes to heat). Assume: 3% efficiency = 3W or 3J/s

X-Rays

• Electromagnetic radiation with short wavelengths – Wavelengths less than for ultraviolet – Wavelengths are typically about 0.1 nm – X-rays have the ability to penetrate most materials with relative ease

• Discovered and named by Roentgen in 1895

• X-rays are produced when high-speed electrons are suddenly slowed down

– Can be caused by the electron striking a metal target

• A current in the filament causes electrons to be emitted

• These freed electrons are accelerated toward a dense metal target

• The target is held at a higher potential than the filament

• The x-ray spectrum has two distinct components • Continuous broad spectrum

– Depends on voltage applied to the tube – Sometimes called bremsstrahlung

• The sharp, intense lines depend on the nature of the target material

• An electron passes near a target nucleus • The electron is deflected from its path by its attraction to the

nucleus – This produces an acceleration

• It will emit electromagnetic radiation when it is accelerated

918

34 8

(3 )(500 10 )8 10

(6.63 10 )(3.0 10 / )

E E J x mN x

hf hc x J s x m s

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Wavelengths Produced

• If the electron loses all of its energy in the collision, the initial energy of the electron is completely transformed into a photon

• The wavelength can be found from

max

min

ƒhc

e V h

• Not all radiation produced is at this wavelength – Many electrons undergo more than one collision before being stopped – This results in the continuous spectrum produced

Example 7: Medical x-ray machines typically operate at a potential difference of 51.00 10x V . Calculate the minimum wavelength their x-ray tubes produce when electrons are accelerated through this potential difference.

Arthur Holly Compton

• 1892 – 1962 • Discovered the Compton effect • Worked with cosmic rays • Director of the lab at U of Chicago • Shared Nobel Prize in 1927

The Compton Effect

• Compton directed a beam of x-rays toward a block of graphite • He found that the scattered x-rays had a slightly longer wavelength that the incident x-rays

– This means they also had less energy • The amount of energy reduction depended on the angle at which the x-rays were scattered • The change in wavelength is called the Compton shift

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min 19 5

(6.63 10 )(3.00 10 / )

(1.60 10 )(1.00 10 )

hc x J s x m s

e V x C x V

111.24 10x m

10

Compton Scattering

• Compton assumed the photons acted like other particles in collisions

• Energy and momentum were conserved • The shift in wavelength is

(1 cos )o

e

h

m c

• The quantity h/mec is called the Compton

wavelength – Compton wavelength = 0.002 43 nm – Very small compared to visible light

• The Compton shift depends on the scattering angle and not on the wavelength

• Experiments confirm the results of Compton scattering and strongly support the photon concept

This is another effect that is correctly predicted by the photon model and not by the wave model.

Photon Interactions; Pair Production

Photons passing through matter can undergo the following interactions: 1. Photoelectric effect: photon is completely absorbed, electron is ejected 2. Photon may be totally absorbed by electron, but not have enough energy to eject it; the

electron moves into an excited state 3. The photon can scatter from an atom and lose some energy 4. The photon can produce an electron-positron pair.

Example 8: X-rays of wavelength 0.200 nm are scattered from a block of material. The scattered

x-rays are observed at an angle of 45o to the incident beam. (a) Calculate the wavelength of the x-rays scattered at this angle.

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31 8

6.63 10(1 cos ) (1 cos45.0 )

(9.11 10 )(3.00 10 / )

o

e

h x J s

m c x kg x m s

137.11 10 0.000711x m nm

0.200711o nm

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Wave-Particle Duality; the Principle of Complementarity

We have phenomena such as diffraction and interference that show that light is a wave, and phenomena such as the photoelectric effect and the Compton effect that show that it is a particle. Which is it? This question has no answer; we must accept the dual wave-particle nature of light.

The principle of complementarity states that both the wave and particle aspects of light are fundamental to its nature. Indeed, waves and particles are just our interpretations of how light behaves.

Louis de Broglie

• 1892 – 1987 • Discovered the wave nature of electrons • Awarded Nobel Prize in 1929

Wave Nature of Matter

Just as light sometimes behaves as a particle, matter sometimes behaves like a wave. The wavelength of a particle of matter is:

h

p

This wavelength is extraordinarily small.

Wave Properties of Particles

• In 1924, Louis de Broglie postulated that because photons have wave and particle characteristics, perhaps all forms of matter have both properties

• Furthermore, the frequency and wavelength of matter waves can be determined

de Broglie Wavelength and Frequency

• The de Broglie wavelength of a particle is h h

p mv

• The frequency of matter waves is ƒE

h

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Example 9: Compare the de Broglie wavelength for an electron 31(9.11 10 )x kg moving at a

speed of 71.00 10 /x m s with that of a baseball of mass 0.145 kg pitched at 45.0 m/s.

Early Models of the Atom It was known that atoms were electrically neutral, but that they could become charged, implying that there were positive and negative charges and that some of them could be removed. One popular atomic model was the “plum-pudding” model: This model had the atom consisting of a bulk positive charge, with negative electrons buried throughout.

Rutherford did an experiment that showed that the positively charged nucleus must be extremely small compared to the rest of the atom. He scattered alpha particles – helium nuclei – from a metal foil and observed the scattering angle. He found that some of the angles were far larger than the plum-pudding model would allow. The only way to account for the large angles was to assume that all the positive charge was contained within a tiny volume – now we know that the radius of the nucleus is 1/10000 that of the atom.

Therefore, Rutherford’s model of the atom is mostly empty space:

3411

31 7

6.63 107.28 10

(9.11 10 )(1.00 10 / )e

e

h x J sx m

m v x kg x m s

34346.63 10

71.02 10(0.145 )(45.0 / )

b

b

h x J sx m

m v kg m s

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Atomic Spectra: Key to the Structure of the Atom

A very thin gas heated in a discharge tube emits light only at characteristic frequencies.

Atomic Spectra: Key to the Structure of the Atom

An atomic spectrum is a line spectrum – only certain frequencies appear. If white light passes through such a gas, it absorbs at those same frequencies.

A portion of the complete spectrum of hydrogen is shown here. The lines cannot be explained by the Rutherford theory. The wavelengths of electrons emitted from hydrogen have a regular pattern: This is called the Balmer series. R is the Rydberg constant: Other series include the Lyman series: And the Paschen series:

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The Bohr Atom Bohr proposed that the possible energy states for atomic electrons were quantized – only certain values were possible. Then the spectrum could be explained as transitions from one level to another. Bohr found that the angular momentum was quantized:

An electron is held in orbit by the Coulomb force: Using the Coulomb force, we can calculate the radii of the orbits:

The lowest energy level is called the ground state; the others are excited states. The correspondence principle applies here as well – when the differences between quantum levels are small compared to the energies, they should be imperceptible.

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de Broglie’s Hypothesis Applied to Atoms De Broglie’s hypothesis is the one associating a wavelength with the momentum of a particle. He proposed that only those orbits where the wave would be a circular standing wave will occur. This yields the same relation that Bohr had proposed. In addition, it makes more reasonable the fact that the electrons do not radiate, as one would otherwise expect from an accelerating charge. These are circular standing waves for n = 2, 3, and 5.